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Full text of "Treatise On Analysis Vol-Ii"

12    THE SPACE L       177

If also F is bounded on [0, 1], then this equality is valid for every integrable
function /on [0, 1]. (Start with the case where F is bounded and /is continuous on
[0, 1 ], then (for/e &*) approximate /by continuous functions. When F is unbounded,
reduce to the case F 2; 0 and approximate F by an increasing sequence of bounded
functions.)

(b)    Deduce from (a) that, for every function / which is integrable on an interval
[a, b] in R, we have

fb                                                 /6

lim      f(t)$inntdt= lim      f(t) cos nt dt =* 0.

n- oo J a                                       n-+oo J a

(c)    Let fp be the canonical mapping of R onto T ~ R/Z, and let //, be the measure
on T which is the image under 9? 1 1 of Lebesgue measure on I; then /x is invariant
under translations in the compact group T(cf. (14.4)). Let k be an integer >1, and let
u : T -> T be the continuous mapping such that u(cp(t)) = u(kt) for /el. The mapping
u is not injective, but ft is invariant with respect to u. Deduce from (a) that u is ergodic
with respect to ju- (use Problem 13(b) of Section 13.9), Deduce that lim sup F(knt)

n-t-oo

is almost everywhere equal to the constant ess sup F(0, and that the same is true of

tel

sup F(/).

nl

6.    Let /e ^g(/x). Show that    (fdfji  = J |/| dp, if and only if there exists c E C with
\c\ = 1 such that/(x) = c|/(#)| almost everywhere.

7.    Suppose that p is bounded and that /x(X) = 1 . Let/e && . If we have f 1 1 + /| dp 2> 1
for every complex number , show that   \fd^ = 0. (For a fixed f e C, consider

8. (a) If lgp<r<+oo, then LrcnL^c:LJnL? and L^nL^cL^nL^. On
the space Uc n L^ , the function Nr,oo = Nr + Nw is a norm with respect to which
the space is complete. Give an example in which the norm induced on L n L by
Np.oo is not equivalent to Nr(CO . (In Lfc n L^ we always have K,oo(/) g 2Np,oo(/).)
The space Uc n L% is a Banach algebra relative to the product fg defined in Section
13.6; it posesses an identity element only if /x is bounded (in which case it is identical
with L).

(b)    The set I(X, //) (or I(/A)) of idempotents in the algebra L^ n L is independent
of r, and consists of the classes <pA  where A runs through the set of integrable subsets
of X. If A, B are integrable subsets of X, the relation cpA = 9B signifies that the set
D(A,B) = AuB-AnB (Section 13.8, Problem 15) is negligible; the relation
/x(D(A, B)) = 0 is an equivalence relation on the set of integrable subsets of X, and
the quotient set may be identified with I(X, ju), We shall write A in place of <pA 
All the norms Nr,oo induce the same topology on I(X, /x), and this topology is therefore
defined by the distance <af(A, S) = Ni(<pA  <PB) = ff,(D(A, B)) with respect to which
I(X, ja) is a complete space. The mappings (A, S)i->-sup(AJ S) (A u B)~ and
(A, B)i-3*inf(A, S)  (A n B)~ are continuous for this topology.

(c)    Let Y be another locally compact space, v a positive measure on Y. If there
exists an isometry U of I(X, ^u) onto I(Y, v) such that U(</>) = $, the measures ft and v
are said to be isometric. For each number p e [1, H-oo [, there then exists a uniqueset of E; let u be a